Finite Group Databasebeta

← #13138066

Subgroups — #13138066

Order 1728.

595 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88648C211yesmaximal, characteristic
2#64101⋊C3C211yesmaximal, characteristic, complemented
3C2×((C3⋊D3)⋊C3)⋊C8C211yesmaximal, characteristic, complemented
4#11720532C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#11720326C211yesmaximal, characteristic, complemented
7#11720318C211yesmaximal, characteristic, complemented
8#10590611—31nomaximal
9#10590610—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
13C2×#6198—21no—
14C2×((C3⋊D3)⋊C3)⋊C4—21no—
15#88648—22no—
16#88648C414yescomplemented
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18C2×#6180C2211yescharacteristic
19#3752⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21C4×(C3⋊Dic3)⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23((C3⋊D3)⋊C3)⋊C8—22no—
24C2×#3799—31no—
25C2×#3796—31no—
26#64101D311yescharacteristic, complemented
27C3×#892—31no—
28C2×#3710—31no—
29C2×D3×C3⋊C8—31no—
30#63636D311yescharacteristic, complemented
31C3×#810—31no—
32C3×#810—31no—
33C3×#810—61no—
34C3×#809—31no—
35#63566—31no—
36#63565—31no—
37#63564—31no—
38#63563—31no—
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44#6229—21no—
45#6229C2×C412yes—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3D411yescharacteristic
48C4⋊((C3⋊D3)⋊C3)—21no—
49C4⋊((C3⋊D3)⋊C3)C2×C412yes—
50C4×(C3⋊D3)⋊C3C2×C412yes—
51C4×(C3⋊D3)⋊C3D412yes—
52#6198—24no—
53((C3⋊D3)⋊C3)⋊C4—24no—
54#6180—21no—
55(C3⋊(C3⋊C8))⋊C3—21no—
56#5002—91no—
57#4804—91no—
58#4772—91no—
59C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×D12—31no—
62C2×C3×C4×D3—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—61no—
66C2×#626—61no—
67C2×#626—61no—
68C2×#624—61no—
69C2×D3×Dic3—61no—
70#3799—34no—
71#3799—62no—
72#3796—34no—
73#3796—62no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4C2×D311yescharacteristic
76C32×C4⋊C4C2×D311yescharacteristic
77#3752—21no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4×C3⋊Dic3—21no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—31no—
83C3×C4×Dic3—61no—
84C2×C3×C3⋊C8—31no—
85C2×C3×C3⋊C8—31no—
86C2×C3×C3⋊C8—31no—
87C2×C3×C3⋊C8—61no—
88#3710—62no—
89D3×C3⋊C8—62no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—21no—
92C2×(C3⋊D3)⋊C3—21no—
93C2×(C3⋊D3)⋊C3—22no—
94C2×(C3⋊D3)⋊C3—41no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3C22⋊C412yes—
97C4×H(3,𝔽3)—41no—
98C4×H(3,𝔽3)C22×C411yescharacteristic
99C4×H(3,𝔽3)C2×D411yescharacteristic
100#1084—22no—
101(C3⋊Dic3)⋊C3—41no—
102(C3⋊Dic3)⋊C3C22⋊C412yes—
103C2×#239—91no—
104C2×#237—91no—
105C2×#235—91no—
106C3×#117—91no—
107C22×C3⋊C8—91no—
108C2×D3×C8—91no—
109C2×D3×C8—91no—
110#892—31no—
111#892—91no—
112C3×#103—91no—
113C3×#102—91no—
114#835—91no—
115#832—91no—
116#821—91no—
117#816—91no—
118#810—31no—
119#810—31no—
120#810—61no—
121#810—91no—
122#809—31no—
123#809—91no—
124#809—91no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3#6711yescharacteristic
127C22×C3×D3—31no—
128C22×C3×D3—61no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3#6711yescharacteristic
137C4⋊(C3⋊D3)—21no—
138C4⋊(C3⋊D3)C4×D312yes—
139C4×C3⋊D3C4×D312yes—
140C4×C3⋊D3D1212yes—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—121no—
146C3×D12—32no—
147C3×D12—61no—
148C3×C4×D3—32no—
149C3×C4×D3—32no—
150C3×Dic6—32no—
151C3×Dic6—32no—
152C3×Dic6—62no—
153C3×Dic6—61no—
154C3×Dic6—61no—
155C3×Dic6—121no—
156#626—64no—
157#626—64no—
158#624—64no—
159D3×Dic3—64no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—61no—
162C3×C3⋊C8—61no—
163C3×C3⋊C8—61no—
164C3×C3⋊C8—121no—
165#416—271noSylow
166C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
167C2×H(3,𝔽3)#13511yescharacteristic
168C2×H(3,𝔽3)#13511yescharacteristic
169(C3⋊D3)⋊C3—22no—
170(C3⋊D3)⋊C3—42no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C22×C3×C4—91no—
174C2×#67—181no—
175C2×#67—181no—
176C22×Dic3—181no—
177#239—94no—
178#239—182no—
179#237—94no—
180#237—182no—
181#235—94no—
182#235—182no—
183C2×D12—31no—
184C2×D12—91no—
185C2×D12—181no—
186C2×C4×D3—31no—
187C2×C4×D3—91no—
188C2×C4×D3—181no—
189C2×C4×D3—181no—
190C2×Dic6—31no—
191C2×Dic6—31no—
192C2×Dic6—61no—
193C2×C3×C8—91no—
194C2×C3×C8—91no—
195C3×C4⋊C4—31no—
196C3×C4⋊C4—31no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—91no—
199C3×C4⋊C4D3211yescharacteristic
200C3×C22⋊C4—181no—
201C3×C42—181no—
202(C2×D3)⋊C4—61no—
203(C2×D3)⋊C4—181no—
204C4⋊Dic3—31no—
205C4⋊Dic3—91no—
206C4×Dic3—61no—
207C4×Dic3—181no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—61no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—182no—
214D3×C8—182no—
215D3×C8—182no—
216C22×C32—21no—
217C22×C32D3×D411yescharacteristic
218C22×C32C2×#6711yescharacteristic
219C2×C3⋊D3—21no—
220C2×C3⋊D3—21no—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—41no—
223C2×C3⋊D3—41no—
224C2×C3⋊D3(C2×D3)⋊C412yes—
225C2×C3×D3—32no—
226C2×C3×D3—61no—
227C2×C3×D3—61no—
228C2×C3×D3—62no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—41no—
234C32×C4—41no—
235C32×C4—81no—
236C32×C4C2×C4×D311yescharacteristic
237C32×C4C2×C4×D311yescharacteristic
238C32×C4C2×D1211yescharacteristic
239C32×C4D3×D411yescharacteristic
240C3⋊Dic3—41no—
241C3⋊Dic3(C2×D3)⋊C412yes—
242C3×Dic3—32no—
243C3×Dic3—62no—
244C3×Dic3—62no—
245C3×Dic3—122no—
246C3×Dic3—121no—
247C2×#41—271no—
248C22×C8—271no—
249#117—91no—
250#103—91no—
251#103—271no—
252#102—91no—
253#102—271no—
254H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
255C23×C3—91no—
256C23×C3—181no—
257C22×D3—31no—
258C22×D3—61no—
259C22×D3—91no—
260C22×D3—181no—
261C3×Q8—92no—
262C3×Q8—181no—
263C3×D4—92no—
264C3×D4—181no—
265C2×C3×C4—21no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—91no—
274C2×C3×C4—121no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4C2×D3211yescharacteristic
278#67—184no—
279#67—184no—
280C2×Dic3—31no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—91no—
285C2×Dic3—121no—
286C2×Dic3—181no—
287C2×Dic3—184no—
288D12—32no—
289D12—61no—
290D12—92no—
291D12—181no—
292D12—184no—
293C4×D3—32no—
294C4×D3—32no—
295C4×D3—92no—
296C4×D3—92no—
297C4×D3—184no—
298C4×D3—184no—
299Dic6—32no—
300Dic6—32no—
301Dic6—62no—
302Dic6—61no—
303Dic6—61no—
304Dic6—121no—
305C3×C8—181no—
306C3×C8—181no—
307C3⋊C8—61no—
308C3⋊C8—61no—
309C3⋊C8—61no—
310C3⋊C8—121no—
311C3⋊C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32D3×C22⋊C411yescharacteristic
316C2×C32#91211yescharacteristic
317C2×C32#91611yescharacteristic
318C2×C32C2×(C2×D3)⋊C411yescharacteristic
319C2×C32#95011yescharacteristic
320C2×C32#95011yescharacteristic
321C3⋊D3—22no—
322C3⋊D3—42no—
323C3×D3—62no—
324C3×D3—122no—
325#41—274no—
326#41—542no—
327C2×Q8—91no—
328C2×D4—91no—
329C2×D4—541no—
330C22×C4—91no—
331C22×C4—541no—
332C2×C8—91no—
333C2×C8—91no—
334C2×C8—542no—
335C4⋊C4—91no—
336C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
337C22⋊C4—181no—
338C42—181no—
339C22×C3—31no—
340C22×C3—31no—
341C22×C3—61no—
342C22×C3—92no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3—182no—
346C22×C3—361no—
347C22×C3—361no—
348C22×C3D3×#6711yescharacteristic
349C2×D3—32no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—62no—
353C2×D3—92no—
354C2×D3—121no—
355C2×D3—121no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—182no—
359C2×D3—361no—
360C2×D3—361no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—31no—
365C3×C4—41no—
366C3×C4—61no—
367C3×C4—61no—
368C3×C4—92no—
369C3×C4—121no—
370C3×C4—121no—
371C3×C4—182no—
372C3×C4—241no—
373C3×C4—361no—
374C3×C4C4×D3211yescharacteristic
375C3×C4D3×D1211yescharacteristic
376Dic3—32no—
377Dic3—62no—
378Dic3—62no—
379Dic3—92no—
380Dic3—122no—
381Dic3—121no—
382Dic3—361no—
383C32—21no—
384C32#477211yescharacteristic, complemented
385C32#500211yescharacteristic, complemented
386C23—91no—
387C23—181no—
388Q8—92no—
389Q8—181no—
390D4—92no—
391D4—181no—
392D4—544no—
393C2×C4—21no—
394C2×C4—91no—
395C2×C4—92no—
396C2×C4—92no—
397C2×C4—181no—
398C2×C4—181no—
399C2×C4—544no—
400C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
401C8—181no—
402C8—181no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—61no—
410C2×C3—61no—
411C2×C3—61no—
412C2×C3—182no—
413C2×C3—362no—
414C2×C3D3×(C2×D3)⋊C411yescharacteristic
415C2×C3#6393311yescharacteristic
416C2×C3#6394711yescharacteristic
417D3—62no—
418D3—122no—
419D3—182no—
420D3—362no—
421C22—92no—
422C22—181no—
423C22—181no—
424C22—182no—
425C22—361no—
426C22—361no—
427C22#8867611yescharacteristic
428C4—41no—
429C4—92no—
430C4—182no—
431C4—361no—
432C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
433C4#8865411yescharacteristic
434C3—31no—
435C3—31no—
436C3—61no—
437C3#1059060811yescharacteristic
438C2—182no—
439C2—362no—
440C2#1172116011yescharacteristic
441C2#1172120111yescharacteristic
442C2#1172123911yescharacteristic
subgroupG/H
centerC22#88676
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×#88648C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#416
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #64101⋊C3C2semidirect
3. C2×((C3⋊D3)⋊C3)⋊C8C2semidirect
4. #11720532C2semidirect
5. #63636⋊C3C2semidirect
6. #11720326C2semidirect
7. #11720318C2semidirect
16. #88648C4semidirect
26. #64101D3semidirect
30. #63636D3semidirect
254. H(3,𝔽3)#416semidirect
384. C32#4772semidirect
385. C32#5002semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#886482
#64101⋊C32
C2×((C3⋊D3)⋊C3)⋊C82
#117205322
#63636⋊C32
#117203262
#117203182
#105906113
#105906103